Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Questions with this tag are about the presentation of a mathematical proof. Questions might include:

  • Should I include [x-mathematical detail] at [y-part of this proof]?
  • Is the following a sufficient proof of [x-mathematical tidbit]?
  • I have written the following proof, could I somehow improve it, does it have good flow/can I improve readability?

But this tag is not for asking someone else to write a proof for you, or for how to answer some question. Questions such as: My professor asked me to prove the Pythagorean theorem and I don't know how to begin are not to have this tag.

This tag is intended for use along with other, more "mathematical" tags. A question about the writing of a proof in abstract algebra, for example, should have as well. This tag can be used along with the proof verification tag.

See here for a useful set of guidelines for writing a solution.

15776 questions
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Proving iff statements, can forward direction just be the reverse of backwards?

Hypothetically, could you just reverse the steps from the forward direction of the proof to do the backward direction? For example, let's say in the forward direction, you take some value $x$ and take the square root to get $\sqrt{x}$, and then add…
amycandyit
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Improve Proof Writing Skills

I've recently delved into proof-writing contests such as the USA(J)MO, and the problems have presented a nice challenge. However, it has been challenging for me to write a neat, concise proof for many of the questions. Does anyone have…
Cogent Coder
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Prove Satisfiability of Property by Set

What is a proof strategy for proving that some property is satisfied by a particular set of numbers. For example, what would be an approach for proving that the archimedean property is satisfied by the rational numbers? In the context I'm coming…
Isaac Kleinman
  • 870
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How do I formulate a proof by equivalences in english?

I have a proof of the form Theorem. $A \iff \forall x D$. Proof. \begin{align} A &\iff \forall x B \\ & \iff \forall x C \\ & \iff \dots \\ & \iff \forall x D \end{align} QED. Note that $A$, $B$, $C$, etc are simply placeholders for more complex…
MyComputer
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Proof regarding multiples of 3

Let $a = x^2 + y^2 + z^2$ for $a, x, y, z \in \mathbb{Z_{>0}}$. I'm trying to prove that if $a$ is divisible by 3, then $x, y, z$ are either ALL divisible by 3, or NONE of them are divisible by 3. I am not sure where to start. I wrote down the…
user843046
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Proof by induction: base case for a limit as x goes to infinity?

With induction we always start with a base case; What would the base case for this be? Choosing 1 seems nonsensical. Choosing infinity seems wrong as well. Prove, using induction, that $\lim\limits_{x\to\infty}\dfrac{(\ln x)^k}x=0$.
MeditationOrBust
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Proof that $\frac{n!}{j_1! j_2! j_3! \cdots j_k!} \in \mathbb{Z}$ if $j_1+ j_2+ j_3+\cdots+ j_k = n$

$n, j_1, j_2, j_3, \ldots, j_k \in \mathbb{N}$ are such that: $j_1+ j_2+ j_3+\cdots+ j_k = n$. Prove that $$\frac{n!}{j_1! j_2! j_3! \cdots j_k!} \in \mathbb{Z}.$$ I don't know how to do it. Tried induction and $e^{\ln(j_1! j_2! j_3! \cdots j_k!)}$…
theboyboy
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Using Archimedean Property to Prove the following

So I've worked through a few of the properties of Archimedas. That is, I understand that for every real number $x$, there exists a natural number $n$ such that $n>x$ And I've also been able to show that, as a consequence of this fact, that for…
Ben Anderson
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If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}, p,q >0$

If $pq=1$, then $p=q=1$ for $p,q \in \mathbb {Z}$, $p,q >0$ I tried to do this by contradiction and I get $(pq=1) \land (p\neq 1 \lor q \neq 1)$ then I have no ideas how to continue with a formal proof. What I know is if i choose q not equal to 1…
Timothy Leung
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If, given a set of nonzero values, the ratio between the min and max is at least $\frac{1}{2}$, is this true for all other values inside the set?

I haven't been able to prove this, or come up with a counterxample. Given a set of nonzero values $A = \{a_1, a_2, ... a_n\}$, and given the hypothesis that ${\min(A)\over\max(A)}\geq\frac{1}{2}$, does the hypothesis imply that such condition is…
Samuele B.
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Proof for Induction

Let $n$ be an integer greater than $0.$ The numbers $1, 2, 3, \ldots, n$ are written on a blackboard. We decide to erase from the blackboard any two numbers, and replace them with their nonnegative difference. After this is done several times, a…
Robert Mugabe
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Need help writing proof

I need help writing a proof for a question from Velleman's "How to Prove It". The question is as follows: Prove that for all real numbers x and y there is a real number z such that x + z = y - z My attempt began by translating the goal into a $P…
Timmy Chyrklund
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How do I prove something is true if $n$ is sufficiently large?

I'm studying for an exam, but am struggling to understand how to prove that something is true if $n$ is sufficiently large. For example, if I'm given $P(n): 2n^3 - 7n^2 \geq 7n -1$, I understand that I need to find an $a$ such that all $n \geq a$…
nicons
  • 63
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$T:P_n(F) \rightarrow F$ PROOF OUTLINE

I'd like some heavy critique if you don't mind. See here for more details. Let $S=\{f \in P_n(F) : f(1)=0\}$. Clearly, the polynomial $f(x)=0 \in S$ because $f(c)=0$ for any choice of $c\in F$. To demonstrate closure under addition and…
Trancot
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Is this an appropriate time to use 'WLOG' in my proof?

A bipartite graph $G=(V,E)$ is known to be connected given its partitions are of equal size $n$, and $|E| \geq n^2 - n + 1$ $G$ is necessarily connected if $|E| \geq n^2 - n + 1$: 1) Based on above $|V|$ = $2n$. 2) WLOG suppose we label the vertices…
PlasticCasio
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